A matrix basis for group algebras of symmetric groups

8 A matrix basis for group algebras of symmetric groups 

In the present section we will give a construction of the matrix basis only for the u = 9VNV operator. The treatment for the other Hermitian operator above is identical and may be supplied by the reader. 

Examining the inner factors of this product, we see that 

All of the coefficients in VNV are real and the matrix M is thus real S5mimetric (and Hermitian). Since the niij are linearly independent, M must be nonsingular. In addition, gp MV is equal to 1, so the diagonal elements of M are all M is essentially an overlap matrix due to the non-orthogonality of the mij. 

We note that if the matrix M were the identity, the would satisfy Eq. (5.20). An 

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